**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## General State of stress at a point

**General State of stress at a point:**** **Stress at a point in a material body has been defined as a force per unit area. But this definition is some what ambiguous since it depends upon what area we consider at that point. Let us, consider a point ‘q' in the interior of the body

Let us pass a cutting plane through a pont 'q' perpendicular to the x - axis as shown below

The corresponding force components can be shown like this

dF

_{x}= s_{xx}. da_{x}dF

_{y}= t_{xy}. da_{x}dF

_{z}= t_{xz}. da_{x}

where da_{x} is the area surrounding the point 'q' when the cutting plane ^ ^{r }is to x - axis. In a similar way it can be assummed that the cutting plane is passed through the point 'q' perpendicular to the y - axis. The corresponding force components are shown below

The corresponding force components may be written as

dF

_{x}= t_{yx}. da_{y}dF

_{y}= s_{yy}. da_{y}dF

_{z}= t_{yz}. da_{y}

where da_{y} is the area surrounding the point 'q' when the cutting plane ^ ^{r }is to y - axis. In the last it can be considered that the cutting plane is passed through the point 'q' perpendicular to the z - axis.

The corresponding force components may be written as

dF

_{x}= t_{zx}. da_{z}dF

_{y}= t_{zy}. da_{z}dF

_{z}= s_{zz}. da_{z}

where da_{z} is the area surrounding the point 'q' when the cutting plane ^ ^{r }is to z - axis. Thus, from the foregoing discussion it is amply clear that there is nothing like __stress at a point 'q'__ rather we have a situation where it is a combination of ** state of stress at a point q.** Thus, it becomes imperative to understand the term state of stress at a point 'q'. Therefore, it becomes easy to express astate of stress by the scheme as discussed earlier, where the stresses on the three mutually perpendiclar planes are labelled in the manner as shown earlier. the state of stress as depicted earlier is called the general or a triaxial state of stress that can exist at any interior point of a loaded body. Before defining the general state of stress at a point. Let us make overselves conversant with the notations for the stresses.

We have already chosen to distinguish between normal and shear stress with the help of symbols s and t .